A note on the applicability of recurrence theorems

We investigate the applicability of Poincaré’s and the quantum recurrence theorems to the physical systems such as a container filled with a gas from a physical perspective. We discuss certain difficulties with regard to the definition of the Hamiltonian for such systems and the existence of one-to-one mappings as solutions to the equations of motion. We examine a typical proof of Poincaré’s recurrence theorem and notice that a key part of the proof appeals to the motion of a neighborhood of finite volume in a phase space and this neighborhood possesses infinite kinetic energy, raising another question concerning the applicability of the theorem to physical systems. The same situations also present themselves in the application of Liouville’s theorem.